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 Polynomials-and-rational-expressions/131295: solve for t: 15t^2-11t=12 a. 4/5 and -3/5 b. 3/4 and -4/3 c. 4/3 and -3/5 d. -4/5 and -3/51 solutions Answer 95873 by jim_thompson5910(28593)   on 2008-03-10 17:00:49 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract 12 from both sides Factor the left side or Set each factor equal to zero or Solve for t in each case --------------------------------- Answer: So the solutions are or So our answer is C
Polynomials-and-rational-expressions/131216: factor completely: 6(y-4)^2-(y-4)-35
1 solutions

Answer 95818 by jim_thompson5910(28593)   on 2008-03-09 23:11:11 (Show Source):
You can put this solution on YOUR website!

Let . So our expression becomes

Plug in

Looking at we can see that the first term is and the last term is where the coefficients are 6 and -35 respectively.

Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient -1? Let's list all of the factors of -210:

Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -210
(1)*(-210)
(2)*(-105)
(3)*(-70)
(5)*(-42)
(6)*(-35)
(7)*(-30)
(10)*(-21)
(14)*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

First NumberSecond NumberSum
1-2101+(-210)=-209
2-1052+(-105)=-103
3-703+(-70)=-67
5-425+(-42)=-37
6-356+(-35)=-29
7-307+(-30)=-23
10-2110+(-21)=-11
14-1514+(-15)=-1
-1210-1+210=209
-2105-2+105=103
-370-3+70=67
-542-5+42=37
-635-6+35=29
-730-7+30=23
-1021-10+21=11
-1415-14+15=1

From this list we can see that 14 and -15 add up to -1 and multiply to -210

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

Now replace z with . Remember, we let

Distribute

Combine like terms

-------------------------------------

So factors to

 Quadratic_Equations/131224: This question is from textbook Algebra II 6X squared + 7X = 3 1 solutions Answer 95817 by jim_thompson5910(28593)   on 2008-03-09 23:05:11 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract 3 from both sides. Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our solutions are or
 Polynomials-and-rational-expressions/131217: find one of the factors of (12x^5y-108x^3y^3) when it is factored completely.1 solutions Answer 95816 by jim_thompson5910(28593)   on 2008-03-09 23:03:53 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor out the GCF Factor to get by using the difference of squares So one of the factors of are , or
 Graphs/131245: -5x+3y=30 x-intercept: y-intercept:1 solutions Answer 95815 by jim_thompson5910(28593)   on 2008-03-09 23:00:13 (Show Source): You can put this solution on YOUR website! Start with the given equation Let's find the x-intercept To find the x-intercept, let y=0 and solve for x: Plug in Simplify Divide both sides by -5 Reduce So the x-intercept is (note: the x-intercept will always have a y-coordinate equal to zero) ------------------ Start with the given equation Now let's find the y-intercept To find the y-intercept, let x=0 and solve for y: Plug in Simplify Divide both sides by 3 Reduce So the y-intercept is (note: the y-intercept will always have a x-coordinate equal to zero) ------------------------------------------ So we have these intercepts: x-intercept: y-intercept:
 Exponents-negative-and-fractional/131243: reduce to lowest terms mn-rs over amn-ars1 solutions Answer 95814 by jim_thompson5910(28593)   on 2008-03-09 22:59:21 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor to get Cancel like terms Simplify -------------------------------------------------- Answer: So simplifies to . In other words
 Polynomials-and-rational-expressions/131244: Given P(x)=2x^5 + 7x^4 - 2x^3 - 25x^2 - 12x + 12 2. Using the DeCartes' rule of signs, state how many positive and how many negative real roots are possible for P(x).(Remember that rational roots are real!)1 solutions Answer 95813 by jim_thompson5910(28593)   on 2008-03-09 22:55:13 (Show Source): You can put this solution on YOUR website!Using Descartes' Rule of Signs, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative) First lets find the number of possible positive real roots: For , simply count the sign changes Here is the list of sign changes: to (positive to negative) to (negative to positive) So there are 2 sign changes, this means there are a maximum of 2 positive roots So there could be 2, or 0 positive real roots Now to find the number of negative real roots, we need to find Replace each x with Simplify Now let's count the sign changes for Here is the list of sign changes: to (negative to positive) to (positive to negative) to (negative to positive) So there are 3 sign changes, this means there are a maximum of 3 negative roots So there could be 3, or 1 negative real roots
 Polynomials-and-rational-expressions/131238: Given P(x)=2x^5 + 7x^4 - 2x^3 - 25x^2 - 12x + 12 1.Using the rational roots theorem, list all possible distinct rational roots of P(x).1 solutions Answer 95811 by jim_thompson5910(28593)   on 2008-03-09 22:44:20 (Show Source): You can put this solution on YOUR website!Any rational zero can be found through this equation where p and q are the factors of the last and first coefficients So let's list the factors of 12 (the last coefficient): Now let's list the factors of 2 (the first coefficient): Now let's divide each factor of the last coefficient by each factor of the first coefficient Now simplify These are all the distinct rational zeros of the function that could occur
 Graphs/131157: I jsut want to be sure I have this problem done right - can you check it for me? Find the slope and y-intercept of the line 8x + 4y = 24. 8x + 4y = 24 - 8x + 4y = -8x + 24 4/4y = -8/4x + 24/4 y = - 2/1x + 6 Thanks!1 solutions Answer 95772 by jim_thompson5910(28593)   on 2008-03-09 15:02:26 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce So the equation is now in slope-intercept form () where the slope is and the y-intercept is So you have the correct answer.
 Inequalities/131142: 2(x-14)-x<7(x+2)+x1 solutions Answer 95764 by jim_thompson5910(28593)   on 2008-03-09 14:21:47 (Show Source): You can put this solution on YOUR website! Start with the given inequality Distribute Combine like terms on the left side Combine like terms on the right side Add 28 to both sides Subtract 8x from both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by -7 to isolate x (note: Remember, dividing both sides by a negative number flips the inequality sign) Divide -------------------------------------------------------------- Answer: So our answer is
 Graphs/131152: y = 9/5x + 21 solutions Answer 95763 by jim_thompson5910(28593)   on 2008-03-09 14:20:55 (Show Source): You can put this solution on YOUR website! Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 9 and the run is 5. This means that to go from point to point, we can go up 9 and over 5 So starting at , go up 9 units and to the right 5 units to get to the next point Now draw a line through these points to graph So this is the graph of through the points and
 Finance/131149: compound interest $500. 5% 10yrs compound quarterly. I know the answer is 821.81 but how. and I know the forumla ia a=p(1+ r/n)^nt help please thank you1 solutions Answer 95762 by jim_thompson5910(28593) on 2008-03-09 14:19:09 (Show Source): You can put this solution on YOUR website! Start with the compound interest formula Plug in p=500, r=0.05 (this is the decimal form of 5% interest), n=4, and t=10 Divide 0.05 by 4 to get 0.0125 Multiply the exponents 4 and 10 to get 40 Add 1 and 0.0125 to get 1.0125 Raise 1.0125 to the 40 th power to get 1.64361946348701 Multiply 500 and 1.64361946348701 to get 821.809731743505 So if you invest$500 at an interest rate of 5%, which is compounded 4 times a year for 10 years, the return is about \$821.81 (which is rounded to the nearest cent)
 Radicals/131144: Please help me! It asks to express irrational solutions in simplest radical form. The problem is (n-5)^2=20 Please and Thank You=)1 solutions Answer 95751 by jim_thompson5910(28593)   on 2008-03-09 12:38:40 (Show Source): You can put this solution on YOUR website! Start with the given equation Take the square root of both sides Simplify the square root (note: If you need help with simplifying the square root, check out this solver) Add 5 to both sides to isolate n. ----------------------------------- Answer: So our solution is: This means our solution breaks down to: or
 Quadratic_Equations/131109: 5) solve 4x+x(x-1)=0 find the x-intercepts of f(x)=4x+x(x-1) What are the solutions x=………. What are the x-intercepts 1 solutions Answer 95734 by jim_thompson5910(28593)   on 2008-03-09 00:33:24 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute Combine like terms Factor the left side Now set each factor equal to zero: or or Now solve for x in each case So our solutions are or which means that the x-intercepts are (-3,0) and (0,0) Notice if we graph we can see that the roots are and . So this visually verifies our answer.
 Quadratic_Equations/131108: 4) A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 60 ft of fence? What should the dimensions of the garden be to give this area?1 solutions Answer 95731 by jim_thompson5910(28593)   on 2008-03-09 00:28:25 (Show Source): You can put this solution on YOUR website!Remember the perimeter of a rectangle is Since one side is formed from the side of the barn, this means that we can take out one length (or width, it doesn't matter) to get Plug in the given perimeter 60 (since he only has 60 ft of fencing) Subtract from both sides Rearrange the equation Now let's introduce another formula. The area of any rectangle is Plug in Rearrange the terms Distribute Rearrange the terms From now on, let's think of as where y is the area and x is the width. Now the equation is in the form of a quadratic which has a vertex that corresponds with the maximum area. So if we find the y-coordinate of the vertex, we can find the max area. In order to find find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex) To find the axis of symmetry, use this formula: From the equation we can see that a=-2 and b=60 Plug in b=60 and a=-2 Multiply 2 and -2 to get -4 Reduce So the axis of symmetry is So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex. Lets evaluate Start with the given polynomial Plug in Raise 15 to the second power to get 225 Multiply 2 by 225 to get 450 Multiply 60 by 15 to get 900 Now combine like terms So the vertex is (15,450) This shows us that the max area is then 450 square feet. So with a width of 15 ft the fence will have a maximum area of 450 square feet Now plug in Multiply Subtract ------------------------------- Answer: So the dimensions of the garden are width: 15, length: 30 Also, the max area of the garden is 450 square feet.
 Quadratic_Equations/131105: Find the vertex and the axis of symmetry for the following graph: 1 solutions Answer 95728 by jim_thompson5910(28593)   on 2008-03-09 00:08:47 (Show Source): You can put this solution on YOUR website!To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex) To find the line of symmetry, use this formula: From the equation we can see that a=-1 and b=2 Plug in b=2 and a=-1 Multiply 2 and -1 to get -2 Reduce So the line of symmetry is So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex. Lets evaluate Start with the given polynomial Plug in Raise 1 to the second power to get 1 Multiply 2 by 1 to get 2 Now combine like terms So the vertex is (1,4) ----------------------- Answer: So the line of symmetry is and the vertex is (1,4) If we graph, we can visually verify our answer Graph of with the line of symmetry and the vertex (1,4)
 Quadratic_Equations/131104: Find the vertex and the axis of symmetry for the following graph: 1 solutions Answer 95727 by jim_thompson5910(28593)   on 2008-03-09 00:01:06 (Show Source): You can put this solution on YOUR website!To find the vertex, we first need to find the line of symmetry (ie the x-coordinate of the vertex) To find the line of symmetry, use this formula: From the equation we can see that a=3 and b=-24 Plug in b=-24 and a=3 Negate -24 to get 24 Multiply 2 and 3 to get 6 Reduce So the line of symmetry is So the x-coordinate of the vertex is . Lets plug this into the equation to find the y-coordinate of the vertex. Lets evaluate Start with the given polynomial Plug in Raise 4 to the second power to get 16 Multiply 3 by 16 to get 48 Multiply 24 by 4 to get 96 Now combine like terms So the vertex is (4,3) ----------------------- Answer: So the line of symmetry is and the vertex is (4,3) If we graph, we can visually verify our answer Graph of with the line of symmetry and the vertex (4,3)
 Quadratic_Equations/131103: 1) A student opens a mathematics book to two facing pages. The product of the page numbers is 420. Find the page numbers. The first page is …………… The second page is ……………. 1 solutions Answer 95726 by jim_thompson5910(28593)   on 2008-03-08 23:56:17 (Show Source): You can put this solution on YOUR website!If we look at the page numbers, we'll notice that they are consecutive integers. So they follow the algebraic form , , , etc. So if the product of two consecutive integers is 420, then we have the equation Distribute Subtract 420 from both sides Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our possible solutions are or However, since a negative page number is not possible, this means that the only solution is Now to find the next page number, simply add 1 to 20 to get So the two page numbers are 20 and 21 Check: To check this answer, simply multiply the two numbers: Works.
 Rational-functions/131061: This question is from textbook simplify 5x+2y/35x+14y1 solutions Answer 95688 by jim_thompson5910(28593)   on 2008-03-08 17:41:58 (Show Source): You can put this solution on YOUR website! Start with the given expression Factor to get Cancel like terms Simplify So simplifies to In other words,
 Inverses/131058: 15=6h 1 solutions Answer 95676 by jim_thompson5910(28593)   on 2008-03-08 16:33:36 (Show Source): You can put this solution on YOUR website! Start with the given equation Subtract 15 from both sides Subtract 6h from both sides Divide both sides by -6 to isolate h Reduce -------------------------------------------------------------- Answer: So our answer is (which is approximately in decimal form)
 Polynomials-and-rational-expressions/131055: I am having trouble with this problem. Can you help me please? Factor by grouping 24x^3+6x^2-4x-1 Thank you so much!1 solutions Answer 95670 by jim_thompson5910(28593)   on 2008-03-08 16:17:11 (Show Source): You can put this solution on YOUR website! Start with the given expression Group like terms Factor out the GCF out of the first group. Factor out the GCF out of the second group Since we have the common term , we can combine like terms So factors to
 Angles/131054: This question is from textbook Right triangle. Length of AB is 2.5. Length of BC is 4.5. B is the right angle. Question: What is Angle A? (This question did not come out of the book listed above but is sample question from a test. The book referenced is a teachers' edition that doesn't go into this kind of detail. But I apparently have to fill in those blanks to get the question posed. Thanks for any help.1 solutions Answer 95666 by jim_thompson5910(28593)   on 2008-03-08 15:55:43 (Show Source): You can put this solution on YOUR website!Let x=measure of angle A If we draw the triangle, it would look something like this: To find angle A, we can use trigonometry to do so. We can use the tangent function to find the measure of angle A Remember So Now take the inverse tangent of both sides Take the inverse tangent of 1.8 to get 60.95 So the measure of angle A is about 60.95 degrees
 expressions/131024: solve for x 4(x+4)=6(y-2)1 solutions Answer 95663 by jim_thompson5910(28593)   on 2008-03-08 15:42:19 (Show Source): You can put this solution on YOUR website! Start with the given equation Distribute the terms on the left side Subtract 16 from both sides Divide both sides by 4 to isolate x
 Linear-systems/131038: Solve the system of equations by the substitution method. 5x+15y=10 2x+8y=6 The solution is: (Type an ordered pair. Type integers or simplified fractions. Type N if there is no solution. Type I if there are infinitely many solutions.)1 solutions Answer 95661 by jim_thompson5910(28593)   on 2008-03-08 15:38:04 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y. So let's isolate y in the first equation Start with the first equation Subtract from both sides Rearrange the equation Divide both sides by Break up the fraction Reduce --------------------- Since , we can now replace each in the second equation with to solve for Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute to Multiply Multiply both sides by the LCM of 3. This will eliminate the fractions (note: if you need help with finding the LCM, check out this solver) Distribute and multiply the LCM to each side Combine like terms on the left side Subtract 16 from both sides Combine like terms on the right side Divide both sides by -2 to isolate x Divide -----------------First Answer------------------------------ So the first part of our answer is: Since we know that we can plug it into the equation (remember we previously solved for in the first equation). Start with the equation where was previously isolated. Plug in Multiply Combine like terms and reduce. (note: if you need help with fractions, check out this solver) -----------------Second Answer------------------------------ So the second part of our answer is: -----------------Summary------------------------------ So our answers are: and which form the point Now let's graph the two equations (if you need help with graphing, check out this solver) From the graph, we can see that the two equations intersect at . This visually verifies our answer. graph of (red) and (green) and the intersection of the lines (blue circle).
 Polynomials-and-rational-expressions/131046: given p(x) = -x^2-x^3+10x, find p(-1)1 solutions Answer 95660 by jim_thompson5910(28593)   on 2008-03-08 15:36:54 (Show Source): You can put this solution on YOUR website! Start with the given function Plug in Raise -1 to the 2nd power to get 1 Multiply -1 and 1 to get -1 Raise -1 to the 3rd power to get -1 Multiply 1 and -1 to get -1 Subtract -1 from -1 to get 0 Multiply 10 and -1 to get -10 Add 0 and -10 to get -10
 Linear-systems/131039: Solve the system of equations by the substitution method. y=6x+8 y=9x+10 The solution is: (Type an ordered pair. Type integers or simplified fractions. Type N if there is no solution. Type I if there are infinitely many solutions.)1 solutions Answer 95657 by jim_thompson5910(28593)   on 2008-03-08 15:34:56 (Show Source): You can put this solution on YOUR website!Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Subtract 10 from both sides Subtract 6x from both sides Combine like terms on the left side Combine like terms on the right side Divide both sides by 3 to isolate x Now that we know that , we can plug this into to find Substitute for each Simplify So our answer is and which also looks like Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer. Graph of (red) and (green)
 Polynomials-and-rational-expressions/131048: when y^3 + 5y^2-3y-15 is factored completely, one of the factors is: 1 solutions Answer 95655 by jim_thompson5910(28593)   on 2008-03-08 15:30:10 (Show Source): You can put this solution on YOUR website! Start with the given expression Group like terms Factor out the GCF out of the first group. Factor out the GCF out of the second group Since we have the common term , we can combine like terms So factors to which means that one of the factors is either or
Polynomials-and-rational-expressions/131049: find one of the factors of: y^2+3y-40
1 solutions

Answer 95653 by jim_thompson5910(28593)   on 2008-03-08 15:28:46 (Show Source):
You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 1 and -40 respectively.

Now multiply the first coefficient 1 and the last coefficient -40 to get -40. Now what two numbers multiply to -40 and add to the middle coefficient 3? Let's list all of the factors of -40:

Factors of -40:
1,2,4,5,8,10,20,40

-1,-2,-4,-5,-8,-10,-20,-40 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -40
(1)*(-40)
(2)*(-20)
(4)*(-10)
(5)*(-8)
(-1)*(40)
(-2)*(20)
(-4)*(10)
(-5)*(8)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3

First NumberSecond NumberSum
1-401+(-40)=-39
2-202+(-20)=-18
4-104+(-10)=-6
5-85+(-8)=-3
-140-1+40=39
-220-2+20=18
-410-4+10=6
-58-5+8=3

From this list we can see that -5 and 8 add up to 3 and multiply to -40

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So factors to which means that one of factors are either or

 Polynomials-and-rational-expressions/131053: Please help me with this problem. Factor 4x(x+6y)+6(x+6y) I got down to 2x(2x+15y+3), but can you factor 2x+15y+3 further? Thank you so much for your help!1 solutions Answer 95652 by jim_thompson5910(28593)   on 2008-03-08 15:27:39 (Show Source): You can put this solution on YOUR website! Start with the given expression Notice how the common factor is . So this means that we can combine like terms. Note: this is very similar to something like Factor the GCF 2 from
Polynomials-and-rational-expressions/131051: factor completely: 6y^3+11y^2-35y
1 solutions

Answer 95651 by jim_thompson5910(28593)   on 2008-03-08 15:25:24 (Show Source):
You can put this solution on YOUR website!

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is where the coefficients are 6 and -35 respectively.

Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient 11? Let's list all of the factors of -210:

Factors of -210:
1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -210
(1)*(-210)
(2)*(-105)
(3)*(-70)
(5)*(-42)
(6)*(-35)
(7)*(-30)
(10)*(-21)
(14)*(-15)
(-1)*(210)
(-2)*(105)
(-3)*(70)
(-5)*(42)
(-6)*(35)
(-7)*(30)
(-10)*(21)
(-14)*(15)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

First NumberSecond NumberSum
1-2101+(-210)=-209
2-1052+(-105)=-103
3-703+(-70)=-67
5-425+(-42)=-37
6-356+(-35)=-29
7-307+(-30)=-23
10-2110+(-21)=-11
14-1514+(-15)=-1
-1210-1+210=209
-2105-2+105=103
-370-3+70=67
-542-5+42=37
-635-6+35=29
-730-7+30=23
-1021-10+21=11
-1415-14+15=1

From this list we can see that -10 and 21 add up to 11 and multiply to -210

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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So our expression goes from and factors further to

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